Question

How do you solve `2x ^ { 3} - x ^ { 2} - 6x < 0`?

Answer 1

`(-oo, -3/2) U (0,2)`

Explanation:

Given: Solve `2x^3 - x^2 - 6x < 0`

Pull out the greatest common factor (GCF): `x(2x^2 - x - 6) < 0`

Now factor the rest of the expression: `x(2x +3)(x-2) < 0`

The function `f(x)=2x^3 - x^2 - 6x =0` has three `x`-intercepts:

`x = 0, x = -3/2, x = 2`

To solve we need to set up intervals and test a value of `x` in the interval to see if it is < 0:

Intervals: `(-oo, -3/2), (-3/2, 0), (0, 2), (2, oo)`
test value: `x = -2, " " x = -1, " " x = 1, " " x = 3`

`f(-2) = 2(-2)^3 - (-2)^2 - 6(-2) = -8 < 0`

`f(-1) = 2(-1)^3 - (-1)^2 - 6(-1) = 3 > 0`

`f(1) = 2(1)^3 - (1)^2 - 6(1) = -5 < 0`

`f(3) = 2(3)^3 - (3)^2 - 6(3) = 27 > 0`

The intervals that are < 0 are:`(-oo, -3/2) U (0, 2)`